Eigenvalue Asymptotics Near a Flat Band in the Presence of a Slowly Decaying Potential

Abstract

We provide eigenvalue asymptotics for a Dirac-type operator on $Z^{n}$ , $n ⩾ 2$ , perturbed by multiplication operators that decay as $| μ |^{- γ}$ with $γ < n$ . We show that the eigenvalues accumulate near the value of the flat band at a “semiclassical” rate with a constant that encodes the structure of the flat band. Similarly, we show that this behavior can be obtained also for a Laplace operator on a periodic graph.

Keywords

Discrete Dirac operator eigenvalue asymptotics flat band

1. Introduction

In this article, we consider an operator whose decomposition into a direct integral features a flat band. We are interested in the accumulation of eigenvalues near the value of the flat band when a perturbation is added. We start by briefly explaining the setting and then discuss our motivation for conducting such an analysis.

Let us denote by $X$ the standard graph structure in $Z^{n}$ and consider the Dirac-type operator on $ℓ^{2} (X)$ defined by the following equation:

H_{0} = (\begin{matrix} m & d^{*} \\ d & - m \end{matrix}),

where

d

is the discrete version of the exterior derivative and

m

a positive constant. We refer to Section 2.1 for the precise definition but one can readily notice that by construction

H_{0}

satisfies the supersymmetry condition making it into an abstract Dirac operator as by Thaller (1992). Moreover, from the analysis of its band functions, see (8) below, we obtain that the spectrum of

H_{0}

σ (H_{0}) = σ_{ess} (H_{0}) = σ_{ac} (H_{0}) = [- \sqrt{m^{2} + 4 n}, - m] ⋃ [m, \sqrt{m^{2} + 4 n}] .

(1)

An essential observation pertinent to this study is that if

n ⩾ 2

- m

is an embedded infinite-dimensional eigenvalue of

H_{0}

. Throughout this article, we will assume that

m > 0

and

n ⩾ 2

Let us now consider a perturbation by a multiplication operator $V : X \to R$ decaying at infinity. Hence we define

H := H_{0} + V .

(2)

Since

V

is a compact operator,

σ_{ess} (H) = σ_{ess} (H_{0})

. Moreover, since

m > 0

, equality (1) tell us that

(- m, m)

is a gap in the essential spectrum of

H

. Then, for

λ \in (0, m)

, we consider the function

N (λ) = Rank 1_{(- m + λ, 0)} (H),

with

1_{Ω}

being the characteristic function over the Borel set

Ω

. Clearly, this function count the number of eigenvalues of

H

(with multiplicity) on the interval

(- m + λ, 0)

Our primary objective is to analyze the asymptotic behavior of $N$ as $λ ↓ 0$ for a specific class of perturbations that decay slowly at infinity. Further details can be found in Definition 3.1, while our main result is presented in Theorem 3.2.

One motivation for studying $N$ stems from our previous work on the distribution of eigenvalues as presented by Miranda et al. (2023). This article extends our prior research in three significant ways: it encompasses the general $n$ -dimensional scenario, incorporates the potential for non-definite perturbations, and addresses potentials with slower rates of decay at infinity. Moreover, we employ a distinct method to derive the effective Hamiltonian, drawing inspiration from the analysis of eigenvalue distributions for magnetic Schrödinger operators, see Raikov (1990), Iwatsuka and Tamura (1998), and Pushnitski and Rozenblum (2011). This approach yields an effective Hamiltonian with a “typical” structure denoted as $P V P$ , where $P$ is a projection.

Another motivation arises from the recent surge in interest surrounding the study of flat bands in the discrete setting. Unlike the common assumption in the continuous case, periodic Schrödinger operators in periodic graphs often exhibit flat bands, as discussed by Sabri and Youssef (2023). While these configurations have long been studied by the physics community, see Bergholtz Liu (2013), Kollár et al. (2020), and references therein, recent attention from the spectral theory community has also emerged, see for instance, Kerner et al. (2023), Pushnitski and Sobolev (to appear), and Zworski (2024). Remarkably, we demonstrate a striking similarity between the results obtained for our Dirac operator and those for the Laplacian on a specific periodic graph showcasing such a flat band, see Theorem 5.3.

We finish this introduction by briefly describing the structure of the article. In Section 2, we give the precise definition of $H_{0}$ and study its main spectral characteristics. In Section 3, we introduce the class of admissible perturbations and state our main result, which we prove in Section 4. Finally, in Section 5, we show a similar result for the standard graph-Laplacian in a particular $Z^{2}$ -periodic graph.

2. Spectral Theory for a Dirac Operator on

Z^{n}

In this section, we provide the definition of $H_{0}$ taking most notations from Parra (2017), see also Anné and Torki-Hamza (2015), recall its integral decomposition, and show the explicit expression of its resolvent as a fibered operator that will be central to our investigations.

2.1. Discrete Dirac Operator

We denote by $X = (V, A)$ the standard graph structure in $Z^{n}$ . That is, the set of vertices $V$ consists of points $μ \in Z^{n}$ and the set of oriented edges $A$ is composed of pairs $(μ, ν)$ such that $ν = μ \pm δ_{j}$ , where ${δ_{j}}_{j = 1}^{n}$ denotes the canonical basis of $Z^{n}$ . An edge in $A$ is written $e = (μ, ν)$ and its transpose $\bar{e} := (ν, μ)$ . Let us consider the vector spaces of $0$ -cochains $C^{0} (X)$ and $1$ -cochains $C^{1} (X)$ given by the following equation:

C^{0} (X) := {f : V \to C}; C^{1} (X) := {g : A \to C ∣ g (e) = - g (\bar{e})} .

The Hilbert spaces

ℓ^{2} (V)

and

ℓ^{2} (A)

are naturally defined by the inner products of cochains:

⟨ f_{1}, f_{2} ⟩_{0} = \sum_{μ \in V} f_{1} (μ) \bar{f_{2} (μ)}

and

⟨ g_{1}, g_{2} ⟩_{1} = \frac{1}{2} \sum_{e \in A} g_{1} (e) \bar{g_{2} (e)}

, respectively.

The coboundary operator $d : ℓ^{2} (V) \to ℓ^{2} (A)$ is defined by the following equation:

d f (e) := f (ν) - f (μ), for e = (μ, ν) \in A .

(3)

This is the discrete version of the exterior derivative and its adjoint

d^{*} : ℓ^{2} (A) \to ℓ^{2} (V)

is given at each edge by the finite sum

d^{*} g (μ) = \sum_{j = 1}^{n} g (μ, μ + δ_{j}) + g (μ, μ - δ_{j}), for μ \in V .

(4)

Let us define the Hilbert space

ℓ^{2} (X) = ℓ^{2} (V) \oplus ℓ^{2} (A)

and denote by

P_{V}

and

P_{A}

the corresponding projections. Further, we introduce the involution

τ

ℓ^{2} (X)

τ (f, g) = (f, - g) .

Then, for a strictly positive constant

m

let us consider the free Dirac operator

H_{0} = d + d^{*} + m τ = (\begin{matrix} 0 & d^{*} \\ d & 0 \end{matrix}) + m (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}) = (\begin{matrix} m & d^{*} \\ d & - m \end{matrix}),

where we have slightly abused notation by considering

d

and

d^{*}

acting on

ℓ^{2} (X)

. Note that

H_{0}

is a Dirac-type operator in the sense that

H_{0}^{2} = (\begin{matrix} Δ_{0} + m^{2} & 0 \\ 0 & Δ_{1} + m^{2} \end{matrix}),

where

Δ_{0}

is the Laplacian on vertices and

Δ_{1}

is the (1-down) Laplacian on edges.

2.2. Integral Decomposition

Let us denote by $H := L^{2} (T^{n}, C^{n + 1})$ . In this section, we construct a unitary operator $U : l^{2} (X) \to H$ . Consider the action of $Z^{n}$ on $X$ given for $μ \in Z^{n}$ , $x \in V$ , and $e = (x, y) \in A$ by

μ x = μ + x and μ e := (μ + x, μ + y) .

Then, a natural class of representatives of the orbits of such action is given by

0 \in V

together with the edges

e_{j} = (0, δ_{j})

and

e_{j}^{-} = (0, - δ_{j})

Let us denote $T^{n} = R^{n} / [0, 1]^{n}$ and set $C_{c} (X)$ to be the set of cochains with compact support, that is, $f \in C_{c} (X)$ if and only if it vanishes except for a finite number of vertices and edges. We define $U : C_{c} (X) \to L^{2} (T^{n}, C^{n + 1})$ by setting, for $f \in C_{c} (X)$ and $ξ \in T^{n}$ ,

(U f) (ξ) = (\sum_{μ \in Z^{n}} e^{- 2 π i ξ \cdot μ} f (μ), \sum_{μ \in Z^{n}} e^{- 2 π i ξ \cdot μ} f (μ e_{1}), \dots, \sum_{μ \in Z^{n}} e^{- 2 π i ξ \cdot μ} f (μ e_{n})) .

Then

U

extends to a unitary operator, still denoted by

U

, from

ℓ^{2} (X)

H

. Further, we set

H_{V} := U P_{V} (ℓ^{2} (X)) ≅ L^{2} (T^{n})

and

H_{A} := U P_{A} (ℓ^{2} (X)) ≅ L^{2} (T^{n}, C^{n})

We draw the reader’s attention to the fact that this definition of $U$ correspond to the following choice of the Fourier transform in $Z^{n}$ :

F : l^{2} (Z^{n}) \to L^{2} (T^{n}); (F f) (ξ) := \sum_{μ \in Z^{n}} e^{- 2 π i ξ \cdot μ} f (μ) .

Finally, let us define the functions

a_{j} (ξ) := - 1 + e^{- 2 π i ξ_{j}} .

The following proposition shows that through conjugation by

U

, the operator

H_{0}

becomes a multiplication operator, enabling the study of its spectral properties through the examination of characteristics of its band functions.

Proposition 2.1 Parra, 2017, Proposition 3.5

The operator $H_{0}$ satisfy that $U H_{0} U^{*} = h_{0}$ , where $h_{0}$ denotes the multiplication operator by the real analytic function $h_{0} : T^{n} \to M_{n + 1 \times n + 1} (C)$ on $L^{2} (T^{n}, C^{n + 1})$ given by the following equation:

h_{0} (ξ) = (\begin{matrix} m & a_{1} (ξ) & \dots & a_{n} (ξ) \\ \bar{a_{1} (ξ)} & - m & \dots & 0 \\ ⋮ & ⋱ & ⋮ \\ \bar{a_{n} (ξ)} & 0 & \dots & - m \end{matrix}) .

(5)

2.3. Spectrum and Resolvent of

H_{0}

The band functions of $h_{0}$ have an explicit expression so we are able to compute $σ (H_{0}) = ⋃_{j} λ_{j} (T^{n})$ . Indeed, from (5) one can see that for $ξ \in T^{n}$ the characteristic polynomial associated to $h_{0} (ξ)$ is given by the following equation:

p_{z} (ξ) = (- 1)^{n} (m + z)^{n - 1} (m^{2} - z^{2} + \sum_{j = 1}^{n} | a_{i} (ξ) |^{2}) .

(6)

(see Lemma A.1).

For convenience, we define $r : T^{n} \to R^{+}$ and $r_{i} : T^{n} \to R^{+}$ for $i \in {1, \dots, n}$ by

r (ξ) := \sum_{j = 1}^{n} | a_{j} (ξ) |^{2} and r_{i} (ξ) = r (ξ) - | a_{i} (ξ) |^{2} = \sum_{j \neq i} | a_{j} |^{2} .

(7)

Thus, there are three band functions:

z_{0} (ξ) = - m, z_{\pm} (ξ) = \pm \sqrt{m^{2} + r (ξ)} .

(8)

From the identities

| a_{j} (ξ) |^{2} = 2 (1 - \cos (2 π ξ_{j})) = 4 \sin^{2} (π ξ_{j}),

(9)

we easily see that the spectrum of

H_{0}

satisfies (1). Moreover, as shown in Figure 1, we can observe that the threshold

- m

correspond to both the maximum of

z_{-}

and the constant value of the flat band. Note from (9) and (8) that

z_{-}

attains its maximum only at

0 \in T^{n}

for every

n

and hence Figure 1 is generic.

Figure 1.

Two views of the three band functions for $n = 2$ . The negative band and the flat band only touch at $(0, 0)$ .

From (5) and for $z \notin σ (H_{0})$ one can check that

\begin{aligned} (h_{0} - z)^{- 1} = & \frac{(- 1)^{n}}{p_{z}} \\ \times (\begin{matrix} (z + m)^{n} & a_{1} (z + m)^{n - 1} & \dots & a_{n} (z + m)^{n - 1} \\ \bar{a_{1}} (z + m)^{n - 1} & (z + m)^{n - 2} (z^{2} - m^{2} - r_{1}) & \dots & a_{n} \bar{a_{1}} (z + m)^{n - 2} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ \bar{a_{n}} (z + m)^{n - 1} & a_{1} \bar{a_{n}} (z + m)^{n - 2} & \dots & (z + m)^{n - 2} (z^{2} - m^{2} - r_{n}) \end{matrix}) \end{aligned}

from where one can obtain

(h_{0} - z)^{- 1} = \frac{1}{(m + z) (m^{2} - z^{2} + r)} (\begin{matrix} (z + m)^{2} & a_{1} (z + m) & \dots & a_{n} (z + m) \\ \bar{a_{1}} (z + m) & z^{2} - m^{2} - r_{1} & \dots & a_{n} \bar{a_{1}} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ \bar{a_{n}} (z + m) & a_{1} \bar{a_{n}} & \dots & z^{2} - m^{2} - r_{n} \end{matrix}) .

(10)

This formula is fully justified in Lemma A.2.

Notice that from the particular form of $h_{0} (ξ) + m$ , that can be obtained directly from (5), we can check that $Ker (h_{0} + m) \subset H_{A}$ . Indeed, one can prove directly that $Ker (H_{0} + m) \subset {0} \times ℓ^{2} (A)$ by constructing for each $μ \in Z^{n}$ a closed path over which we define a cochain alternating the values $1$ and $- 1$ , see Miranda et al. (2023, Section 2) for an explicit construction for the $Z^{2}$ case. We stress that the flat bands of discrete periodic graphs are known to be associated with finitely supported eigenfunctions (Kuchment, 1991).

3. Perturbed Operator and Main Result

We turn now our attention to the concrete class of perturbations $V$ that we will treat in this article. A symmetric multiplication operator on $ℓ^{2} (X)$ is defined by $V : X \to R$ such that $V (e) = V (\bar{e})$ for every $e \in A$ . Given such a $V$ , our full Hamiltonian is defined by (2).

Further, we define the following real-valued functions on $Z^{n}$

v_{0} (μ) := V (μ); v_{j} (μ) := V (μ e_{j}), 1 ⩽ j ⩽ n .

(11)

This choice allows us to further specify the decay of

V

at infinity, but other choices of representatives would give the same type of decay. Let us consider the class of symbols

S^{γ} (Z^{n})

given by the functions

v : Z^{n} \to C

that satisfies for any multi-index

α = (α_{1}, \dots, α_{n}) \in N^{n}

| D^{α} v (μ) | ⩽ C_{α} ⟨ μ ⟩^{- γ - | α |},

(12)

where

D_{j} v (μ) := v (μ + δ_{j}) - v (μ)

| α | := \sum_{j = 1}^{n} α_{j}

, and

D^{α} := D_{1}^{α_{1}} \dots D_{n}^{α_{n}}

Definition 3.1

We call a perturbation $V$ admissible of order $γ$ , with $n > γ > 0$ , if ${v_{j}}_{j = 0}^{n} \in S^{γ} (Z^{n})$ and for $j = 1, \dots, n$

v_{j} (μ) = ⟨ μ ⟩^{- γ} (Γ_{j} + o (1)) as μ \to \infty,

(13)

with

Γ_{j} \neq 0

for at least one

j

This condition may appear restrictive, but it simplifies the presentation of the results. Naturally, alternative classes of symbols and asymptotic behaviors at infinity of the $v_{j}$ ’s could be addressed using akin methods to those employed in this article.

For an admissible perturbation, we define the diagonal $(n + 1) \times (n + 1)$ matrix $Γ$ by the following equation:

Γ_{l l} = {\begin{cases} Γ_{l - 1} & if Γ_{l - 1} > 0, \\ 0 & otherwise . \end{cases}

(14)

We define as well the function

M : T^{n} \to M_{(n + 1) \times (n + 1)} (C)

by the following equation:

M (ξ) := \frac{1}{r (ξ)} (\begin{matrix} 0 & 0 & 0 & \dots & 0 \\ 0 & r_{1} (ξ) & - a_{2} (ξ) \bar{a_{1} (ξ)} & \dots & - a_{n} (ξ) \bar{a_{1} (ξ)} \\ 0 & - a_{1} (ξ) \bar{a_{2} (ξ)} & r_{2} (ξ) & \dots & - a_{n} (ξ) \bar{a_{2} (ξ)} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & - a_{1} (ξ) \bar{a_{n} (ξ)} & - a_{2} (ξ) \bar{a_{n} (ξ)} & \dots & r_{n} (ξ) \end{matrix}) .

(15)

Theorem 3.2

Assume that $V$ is an admissible perturbation of order $γ$ . Define the constant $C$ by

C := \int_{T^{n}} Tr ((Γ M (ξ))^{\frac{n}{γ}}) d ξ .

(16)

Let

τ_{n}

denotes the volume of the unitary sphere in

R^{n}

. Then, the eigenvalue counting function satisfies

N (λ) = λ^{- \frac{n}{γ}} (C τ_{n} + o (1)), λ ↓ 0.

(17)

Remark 3.3

The best-known case of degenerate eigenvalues in the continuous setting is the Landau Hamiltonian on $R^{2}$ . Although they are not usually thought of as flat bands, the direct integral decomposition obtained from the Landau gauge gives us that each Landau level is the image of a constant band function in $R$ . In this sense, it is somewhat natural that the asymptotic order obtained in (17) coincides with the result by Raikov (1990, Theorem 2.6), see also (35). However, the constants differ in both cases. For the Landau Hamiltonian, the constant depends only on the multiplicity of the corresponding Landau level and the intensity of the magnetic field, whereas for the discrete Dirac operator, the perturbation interacts with the associated eigenspace non-trivially as encoded by $C$ .

4. Proof

In this section, we will prove our main result in Theorem 3.2. Before that, we start by recalling some known results on compact operators in order to settle notation and then reduce the study of $N$ to the study of the eigenvalue counting function of an effective Hamiltonian.

4.1. Some Notation and Results on Compact Operators

Given the Hilbert spaces $H_{1}$ and $H_{2}$ , we denote by $S_{\infty} (H_{1}, H_{2})$ the class of compact operators from $H_{1}$ to $H_{2}$ . When $H_{1} = H_{2} = H$ , we will just write $S_{\infty} (H)$ . For $K = K^{*} \in S_{\infty} (H)$ and $s > 0$ , we set

n_{\pm} (s; K) := Rank 1_{(s, \infty)} (\pm K) .

Thus, the functions

n_{\pm} (\cdot; K)

are, respectively, the counting functions of the positive and negative eigenvalues of the operator

K

. For

K \in S_{\infty} (H_{1}, H_{2})

, we define

n_{*} (s; K) := n_{+} (s^{2}; K^{*} K), s > 0;

thus

n_{*} (\cdot; K)

is the counting function of the singular values of

K

which, when ordered non-increasingly, we denote by

{s_{j} (K)}

. Let

K_{j}

j = 1, 2

be self-adjoint compact operators. For

s_{1}, s_{2} > 0

, we have the Weyl inequalities (see e.g., Birman & Solomjak, 1987, Theorem 9.2.9)

n_{\pm} (s_{1} + s_{2}; K_{1} + K_{2}) ⩽ n_{\pm} (s_{1}; K_{1}) + n_{\pm} (s_{2}; K_{2}) .

(18)

If instead we only have

{K_{1}, K_{2}} \subset S_{\infty} (H_{1}, H_{2})

, the Ky Fan inequality (see e.g., Birman & Solomjak, 1987, Subsection 11.1.3) gives

n_{*} (s_{1} + s_{2}; K_{1} + K_{2}) ⩽ n_{*} (s_{1}; K_{1}) + n_{*} (s_{2}; K_{2}) .

(19)

Further, for

0 < p < \infty

, we define the class of compact operators

S_{p, w}

S_{p, w} := {K \in S_{\infty} : s_{j} (K) = O (j^{- 1 / p})},

together with the quasi-norm

| | K | |_{p, w} := sup_{j} {j^{1 / p} s_{j} (K)} = {(sup_{s > 0} {s^{p} n_{*} (s; K))}^{1 / p},

that satisfies the “weakened triangle inequality”

| | K_{1} + K_{2} | |_{p, w} ⩽ 2^{1 / p} (| | K | |_{p, w} + | | K | |_{p, w}),

and the ‘‘weakened Hölder inequality”

| | K_{1} K_{2} | |_{r, w} ⩽ c (p, q) | | K_{1} | |_{p, w} | | K_{2} | |_{q, w},

(20)

for

r^{- 1} = p^{- 1} + q^{- 1}

and

c (p, q) = (p / r)^{1 / p} (q / r)^{1 / q}

(see Birman & Solomjak, 1987, Chapter 11).

Finally, consider the set $l_{p, w}$ of functions $v : Z^{n} \to C$ such that

# {μ \in Z^{n} : | v (μ) | > λ} = O (λ^{- p}) .

Let us finish this section by considering the following result, which is a particular case of Birman et al. (1991, Theorem 4.8(ii)).

Proposition 4.1 Cwikel-Birman-Solomyak

Let $p > 2$ and assume $v \in l_{p, w}$ and $f \in L^{p} (T^{n})$ . Then $f F v \in S_{p, w} (ℓ^{2} (Z^{n}), L^{2} (T^{n}))$ , and there exists a positive constant $C (n)$ such that

| | f F v | |_{p, w} ⩽ C (n) ‖ v ‖_{l_{p, w}} ‖ f ‖_{L^{p} (T^{n})} .

4.2. Effective Hamiltomian

In this section, we will use the notation

N ((a, b); T) := Rank 1_{(a, b)} (T),

where

a < b

and T is a self-adjoint operator without essential spectrum in

(a, b)

. Following the approach coming from the study of magnetic Schrödinger operators, our aim is to study

P V P

, where

P

stands for the projection on the flat band, that is,

P := 1_{{- m}} (H_{0}) .

(21)

Lemma 4.2

Recall that $M : T^{n} \to M_{(n + 1) \times (n + 1)}$ was defined in (15). Then

P = U^{*} M U .

Proof.

By Stone formula one can check that

P = {s - \lim}_{κ ↓ 0} \frac{1}{i π} \int_{- m}^{0} ((H_{0} - s - i κ)^{- 1} - (H_{0} - s + i κ)^{- 1}) d s .

Then, from (10) we need only to check that

lim_{κ ↓ 0} \int_{- m}^{0} (\frac{1}{m + λ + i κ} - \frac{1}{m + λ - i κ}) d λ = - i π δ_{- m},

where

δ_{- m}

is the Dirac delta function on

- m

. □

Now, set $P^{⊥} := Id - P$ and for $κ > 0$ define

H_{κ}^{\pm} := H_{0} + P (V \pm κ | V |) P + P^{⊥} (V \pm κ^{- 1} | V |) P^{⊥}

Then, by Pushnitski and Rozenblum (2011, Lemma 4.2)

H_{κ}^{-} ⩽ H ⩽ H_{κ}^{+} .

(22)

Then, arguing as in the proof of Pushnitski and Rozenblum (2011, Theorem 4.1(ii)), we obtain that:

\begin{aligned} \pm N (λ) ⩽ & \pm N ((- m + λ, 0); - m P + P (V \pm κ | V |) P) \\ \pm N ((- m + λ, 0); P^{⊥} (H_{0} + (V \pm κ^{- 1} | V |)) P^{⊥}) + O (1) . \end{aligned}

(23)

In the next lemma, we treat the second term on the right-hand side of (23), showing that the perturbation

V

interacts with the complement of the degenerated eigenspace only at a lesser order.

Lemma 4.3

N ((- m + λ, 0); P^{⊥} (H_{0} + V \pm κ^{- 1} | V |) P^{⊥}) = o (λ^{- n / γ}), λ ↓ 0.

Proof.

Define the function $W : X \to R^{+}$ by $w_{j} (μ) = ⟨ μ ⟩^{- γ}$ , where we are using the notation of (11). From (12), there exist a constant $C > 0$ such that $| V | ⩽ C W$ . Denote by $W_{κ} := C (1 + κ^{- 1}) W$ . Then it can been seen that (again as in the proof of Pushnitski & Rozenblum, 2011, Theorem 4.1(ii))

N ((- m + λ, 0); P^{⊥} (H_{0} + V \pm κ^{- 1} | V |) P^{⊥}) ⩽ N ((- m + λ, 0); P^{⊥} (H_{0} + W_{κ}) P^{⊥}) + O (1) .

Now, by the Birman-Schwinger principle (see, for instance, Klaus, 1982; Pushnitski, 2009), we get for

λ \in (0, m)

N ((- m + λ, 0); P^{⊥} (H_{0} + W_{κ}) P^{⊥}) = n_{+} (1; P^{⊥} W_{κ}^{1 / 2} P^{⊥} (H_{0} + m - λ)^{- 1} P^{⊥} W_{κ}^{1 / 2} P^{⊥}) + O (1) .

Define the

(n + 1) \times (n + 1)

matrix

M_{R} := (\begin{matrix} λ & a_{1} & \dots & a_{n} \\ \bar{a_{1}} & λ - 2 m & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ \bar{a_{n}} & 0 & \dots & λ - 2 m \end{matrix}) .

Then, from (10) and Lemma 4.2, it is not difficult to see that for

λ \in (0, m)

(H_{0} + m - λ)^{- 1} P^{⊥} = U^{*} \frac{M_{R}}{r + λ (2 m - λ)} (Id - M) U,

where

Id

denotes the identity

(n + 1) \times (n + 1)

matrix. Furthermore, the operator

W_{κ}^{1 / 2} P^{⊥} (H_{0} + m - λ)^{- 1} P^{⊥} W_{κ}^{1 / 2}

is obviously compact and from (20)

\begin{aligned} ‖ W_{κ}^{1 / 2} P^{⊥} (H_{0} + m - λ)^{- 1} P^{⊥} W_{κ}^{1 / 2} ‖_{n / γ, w} & ⩽ C ‖ W_{κ}^{1 / 2} U^{*} \frac{1}{r + λ (2 m - λ)} ‖_{2 n / γ, w} \\ \times ‖ M_{R} (Id - M) U W_{κ}^{1 / 2} ‖_{2 n / γ, w} . \end{aligned}

Consider the operator

W_{κ}^{1 / 2} U^{*} 1 / r + λ (2 m - λ)

. Since

1 / r + λ (2 m - λ)

is bounded, it is in

L^{p} (T^{n})

for any

p > 1

. Further, each component of the multiplication operator

W_{κ}^{1 / 2}

is in

l_{2 n / γ, w}

. Then, since

2 n / γ > 2

, by Proposition 4.1,

{‖ W_{κ}^{1 / 2} U^{*} \frac{1}{r + λ (2 m - λ)} ‖}_{2 n / γ, w} ⩽ C {‖ \frac{1}{r + λ (2 m - λ)} ‖}_{L^{\frac{2 n}{γ}} (T^{n})} ‖ W_{κ}^{1 / 2} ‖_{\oplus_{l = 0}^{n} l_{2 n / γ}, w} .

To estimate the

L^{\frac{2 n}{γ}}

norm, we use the coarea formula

\begin{aligned} \int_{T^{n}} | (r + λ (2 m - λ))^{- 1} |^{2 n / γ} = & \int_{0}^{1 / 2} \frac{1}{(ρ + λ (2 m - λ))^{2 n / γ}} \int_{r (ξ) = ρ} \frac{1}{| \nabla r (ξ) |} d ξ d ρ \\ ⩽ & C \int_{0}^{1 / 2} \frac{1}{(ρ + λ (2 m - λ))^{2 n / γ}} \int_{r (ξ) = ρ} \frac{1}{r (ξ)^{1 / 2}} d ξ d ρ \\ ⩽ & C \int_{0}^{1 / 2} \frac{1}{(ρ + λ (2 m - λ))^{2 n / γ}} \frac{1}{ρ^{1 / 2}} \int_{r (ξ) = ρ} d ξ d ρ \\ ⩽ & C \int_{0}^{1 / 2} \frac{ρ^{n / 2 - 1}}{(ρ + λ (2 m - λ))^{2 n / γ}} d ρ \\ ⩽ & C λ^{- 2 n / γ + n / 2}, \end{aligned}

where in the first and third inequalities, we have used (9). Analogously,

{‖ M_{R} (Id - M) U W_{κ}^{1 / 2} ‖}_{2 n / γ, w} ⩽ C ‖ W_{κ}^{1 / 2} ‖_{\oplus_{l = 0}^{n} l_{2 n / γ}, w},

since the matrix

M_{R} (Id - M)

is bounded with uniform bound in

λ

. Putting all this together we obtain

‖ W_{κ}^{1 / 2} P^{⊥} (H_{0} + m - λ)^{- 1} P^{⊥} W_{κ}^{1 / 2} ‖_{n / γ, w} ⩽ C λ^{γ / 4 - 1},

which is equivalent to say that

\begin{aligned} n_{*} (s; W_{κ}^{1 / 2} P^{⊥} (H_{0} + m - λ)^{- 1} P^{⊥} W_{κ}^{1 / 2}) ⩽ & C λ^{- n / γ + n / 4} \\ = & o (λ^{- n / γ}) . ◼ \end{aligned}

□

4.3. Eigenvalue Counting Function for the Effective Hamiltonian

From (23) and Lemma 4.3

\pm N (λ) ⩽ \pm N ((λ, m); P (V \pm κ | V |) P) + o (λ^{- n / γ}), λ ↓ 0.

(24)

Then, we are led to study the distribution of positives eigenvalues of the compact operator

P (V \pm κ | V |) P

For ease of notation, for any $κ > 0$ we define $T_{κ}^{\pm}$ in $S_{\infty} (H)$ by

T_{κ}^{\pm} := U P (V \pm κ | V |) P U^{*} = M U (V \pm κ | V |) U^{*} M .

Proposition 4.4

For an admisible $V$ , we have

n_{+} (λ; T_{κ}^{\pm}) = (\frac{1 \pm κ}{λ})^{n / γ} τ_{n} \int_{T^{n}} Tr ((M (ξ) Γ M (ξ))^{n / γ}) d ξ (1 + o (1)), λ ↓ 0.

In order to prove this proposition, we follow the ideas of Miranda et al. (2023, Theorem 6.1), which in turn are inspired by the proof of Birman and Solomjak (1970, Theorem 1). By analogy, we denote $◻ := [0, 1)^{n} \subset R^{n}$ and hence $H ≅ \oplus_{j = 0}^{n} L^{2} (◻)$ . Finally, for ease of notation, let us set ${\hat{V}}_{κ}^{\pm} = U (V \pm κ | V |) U^{*}$ .

Remark 4.5

The statement of Proposition 4.4 is particular to our effective Hamiltonian and problem. However, in the proof we only use that $M \in L^{p} (T^{d}; C^{n + 1})$ for $p > 2$ and we could also replace $V \pm κ | V |$ with another potential satisfying (12) and (13). A similar statement holds for $n_{-}$ .

Lemma 4.6

Let $X$ and $Y$ be two subsets of $◻$ with no interior points in common. Then

n_{*} (r; 1_{X} {\hat{V}}_{κ}^{\pm} 1_{Y}) = o (r^{- n / γ}), r ↓ 0.

Proof.

The proof uses Proposition 4.1 and is almost equal to the proof of Miranda et al. (2023, Lemma 6.4). □

Lemma 4.7

Let ${◻_{j}}$ be a partition of $◻$ into cubes of equal size $1 / q^{n}$ , $q \in Z_{+}$ , and let ${B_{j}}_{j = 1}^{q^{n}}$ be matrices in $M_{(n + 1) \times (n + 1)} (C)$ . Let ${\overset{ˇ}{T}}_{κ}^{\pm} : \oplus_{j = 0}^{n} L^{2} (◻) \to \oplus_{j = 0}^{n} L^{2} (◻)$ be the operator defined by the following equation:

{\overset{ˇ}{T}}_{κ}^{\pm} = \sum_{j} B_{j} 1_{◻_{j}} {\hat{V}}_{κ}^{\pm} 1_{◻_{j}} B_{j}^{*} .

Then, for any

δ \in (0, 1)

\begin{aligned} \frac{τ_{n}}{q^{n}} \sum_{j} Tr & ((B_{j} Γ (1 \pm κ) B_{j}^{*})^{n / γ}) (λ (1 + δ))^{- n / γ} (1 + o (1)) \\ ⩽ & n_{+} (λ; {\overset{ˇ}{T}}_{κ}^{\pm}) \\ ⩽ & \frac{τ_{n}}{q^{n}} \sum_{j} Tr ((B_{j} Γ (1 \pm κ) B_{j}^{*})^{n / γ}) (λ (1 - δ))^{- n / γ} (1 + o (1)), λ ↓ 0. \end{aligned}

Proof.

We will show the proof of the upper bound. The lower bound is similar. Let $B_{0}$ be a constant $(n + 1) \times (n + 1)$ matrix. Then for any $δ \in (0, 1)$

\begin{aligned} n_{+} (λ; B_{0} {\hat{V}}_{κ}^{\pm} B_{0}^{*}) ⩾ & \sum_{j} n_{+} (λ (1 + δ); B_{0} 1_{◻_{j}} {\hat{V}}_{κ}^{\pm} 1_{◻_{j}} B_{0}^{*}) - n_{-} (λ δ; \sum_{j \neq l} B_{0} 1_{◻_{j}} {\hat{V}}_{κ}^{\pm} 1_{◻_{l}} B_{0}^{*}) \\ = & q^{n} n_{+} (λ (1 + δ); B_{0} 1_{◻_{0}} {\hat{V}}_{κ}^{\pm} 1_{◻_{0}} B_{0}^{*}) + o (λ^{- n / γ}), \end{aligned}

where for the inequality we used (18). For the equality, we used first the fact that each operator

B_{0} 1_{◻_{j}} {\hat{V}}_{κ}^{\pm} 1_{◻_{j}} B_{0}^{*}

is unitary equivalent to

B_{0} 1_{◻_{0}} {\hat{V}}_{κ}^{\pm} 1_{◻_{0}} B_{0}^{*}

, for

◻_{0} = (0, 1 / q)^{n}

. Then we used Lemma 4.6 and (19). It follows that:

n_{+} (λ; B_{0} 1_{◻_{0}} {\hat{V}}_{κ}^{\pm} 1_{◻_{0}} B_{0}^{*}) ⩽ \frac{1}{q^{n}} n_{+} (λ (1 - δ); B_{0} {\hat{V}}_{κ}^{\pm} B_{0}^{*}) + o (λ^{- n / γ}) .

(25)

Define

v_{*} (μ) = ⟨ μ ⟩^{- γ}

. One can check that

# {μ \in Z^{n} : v_{*} (μ) > λ} = τ_{n} λ^{- n / γ} (1 + o (1)), λ ↓ 0,

(26)

see, for instance, (Reed & Simon, 1978, Proposition 2 XIII.15).

From (13) set ${\hat{V}}_{*, κ}^{\pm} := U Γ (1 \pm κ) ⟨ μ ⟩^{- γ} U^{*}$ and use the Weyl inequalities (18) to obtain that for $\tilde{δ} \in (0, 1)$

n_{+} (λ; B_{0} {\hat{V}}_{κ}^{\pm} B_{0}^{*}) ⩽ n_{+} (λ (1 - \tilde{δ}); B_{0} {\hat{V}}_{*, κ}^{\pm} B_{0}^{*}) + n_{+} (λ \tilde{δ}; B_{0} ({\hat{V}}_{κ}^{\pm} - {\hat{V}}_{*, κ}^{\pm}) B_{0}^{*}) .

Now, denote by

{β_{0, l}}

the eigenvalues of the matrix

B_{0} Γ B_{0}^{*}

. We have that the eigenvalues of

B_{0} {\hat{V}}_{0}^{\pm} B_{0}^{*}

are given by

{β_{0, l} v_{*} (μ) : 1 ⩽ l ⩽ k, μ \in Z^{d}} .

Thus (26) implies that

\begin{aligned} n_{+} (λ; B_{0} {\hat{V}}_{0}^{\pm} B_{0}^{*}) & = # {1 ⩽ l ⩽ k, μ \in Z^{n} : β_{0, l} v_{*} (μ) > λ} \\ = \sum_{β_{0, l} > 0} n_{+} (\frac{λ}{β_{0, l}}; v_{*}) \\ = (τ_{n} \sum_{β_{0, l} > 0} β_{0, l}^{n / γ}) λ^{- n / γ} (1 + o (1)), λ ↓ 0. \end{aligned}

The same reasoning can be used to show that

n_{+} (B_{0} ({\hat{V}}_{κ}^{\pm} - {\hat{V}}_{0}) B_{0}^{*}) = o (λ^{- n / γ})

. Putting the previous inequalities together, for all

δ, \tilde{δ} \in (0, 1)

\begin{aligned} n_{+} (λ; {\overset{ˇ}{T}}_{κ}^{\pm}) & = \sum_{j} n_{+} (λ; B_{j} 1_{◻_{j}} {\hat{V}}_{κ}^{\pm} 1_{◻_{j}} B_{j}^{*}) \\ ⩽ \frac{1}{q^{n}} \sum_{j} n_{+} (λ (1 - δ); B_{j} {\hat{V}}_{κ}^{\pm} B_{j}^{*}) + o (λ^{- n / γ}) \\ ⩽ \frac{1}{q^{n}} \sum_{j} n_{+} (\frac{λ (1 - δ)}{(1 + \tilde{δ})}; B_{j} {\hat{V}}_{0}^{\pm} B_{j}^{*}) + o (λ^{- n / γ}) \\ = \frac{τ_{n}}{q^{n}} \sum_{j} Tr ((B_{j} Γ (1 \pm κ) B_{j}^{*})^{n / γ}) {(\frac{λ (1 - δ)}{(1 + \tilde{δ})})}^{- n / γ} (1 + o (1)), λ ↓ 0. ◼ \end{aligned}

□

Proof. Proof of Proposition 4.4.

Let $ε > 0$ , and take $B_{ε} = \sum_{j} B_{ε, j} 1_{◻_{ε, j}}$ a step matrix function such that $‖ M - B_{ε} ‖_{L^{p} (T^{n})} < ε$ . Assume that the size of each cube $◻_{ε, j}$ is $1 / q^{n}$ as in the previous lemma.

Take $S_{ε} := B_{ε} {\hat{V}}_{κ}^{\pm} B_{ε}^{*}$ . Then by Proposition 4.1, $‖ T_{κ}^{\pm} - S_{ε} ‖_{n / γ, w} < C ε$ , which means that

n_{*} (λ; T_{κ}^{\pm} - S_{ε}) ⩽ (C ε)^{n / γ} λ^{- n / γ} .

(27)

Also, let

{\overset{ˇ}{T}}_{ε, κ}^{\pm} = \sum_{j} B_{ε, j} 1_{◻_{ε, j}} {\hat{V}}_{κ}^{\pm} B_{ε, j} 1_{◻_{ε, j}}

. Thus, by Lemma 4.6

n_{*} (s; S_{ε} - {\overset{ˇ}{T}}_{ε, κ}^{\pm}) = o (s^{- n / γ}) .

(28)

Now, using Lemma 4.7, we have that for any

δ \in (0, 1)

\begin{aligned} \frac{τ_{n}}{q^{n}} \sum_{j} Tr ((B_{ε, j} & Γ (1 \pm κ) B_{ε, j}^{*})^{n / γ}) (λ (1 + δ))^{- n / γ} (1 + o (1)) \\ ⩽ & n_{+} (λ; {\overset{ˇ}{T}}_{ε, κ}^{\pm}) \\ ⩽ & \frac{τ_{n}}{q^{n}} \sum_{j} Tr ((B_{ε, j} Γ (1 \pm κ) B_{ε, j}^{*})^{n / γ}) (λ (1 - δ))^{- n / γ} (1 + o (1)), λ ↓ 0. \end{aligned}

(29)

Finally, putting together (18), (19), and (27)–(29), and making

λ

δ

, and

ε

goes to

0

, we finish the proof. □

Proof of Theorem 3.2.

The result follows from Proposition 4.4 by taking $κ ↓ 0$ , (24) and using the cyclicity of the trace. □

5. The Laplacian on a Particular

Z^{2}

-Periodic Graph

5.1. A Simple Example of a $Z^{2}$ -Periodic Graph With a Flat Band

Let us start by briefly recalling some notions from the periodic graph theory, we refer to Sunada (2013), Korotyaev and Saburova (2014), and Parra and Richard (2018) for more details. We say that a graph is $Z^{d}$ -periodic if it admits an action of $Z^{d}$ by graph-automorphisms. By fixing representatives of each orbit of vertices for this action, we can define the entire part of a vertex by $⌊ x ⌋ \overset{ˇ}{x} = x$ , where $\overset{ˇ}{x}$ is the representative of the orbit of $x$ . Then, the index of an oriented edge $e = (x, y)$ is just $η (e) = ⌊ y ⌋ - ⌊ x ⌋$ . Note that $η$ is $Z^{d}$ -periodic and hence we can refer to the index of an edge in the quotient graph.

Let us now denote by $\tilde{X} = (\tilde{V}, \tilde{A})$ the graph obtained from $Z^{2}$ by adding a vertex on each edge with trivial weights (see Figure 2(a)). The quotient graph obtained by the action of $Z^{2}$ is composed by three vertices and four edges as presented in Figure 2(b). If we takes as representatives the vertices $(0, 0)$ , $(0, \frac{1}{2})$ , and $(\frac{1}{2}, 0)$ , one can easily check that $η (e_{1}) = η (e_{2}) = (0, 0)$ while $η (e_{3}) = (1, 0)$ and $η (e_{4}) = (0, 1)$ .

Figure 2.

(a) The periodic graph obtained from $Z^{2}$ by adding a vertex to each edge and (b) the quotient graph by the usual action of $Z^{2}$ .

Set ${\tilde{H}}_{0} = - Δ_{0}$ , where $Δ_{0}$ is the usual graph Laplacian, that is, for $f \in ℓ^{2} (\tilde{V})$ and $x \in \tilde{V}$ :

({\tilde{H}}_{0} f) (x) = \sum_{e \in \tilde{A}, e = (x, y)} f (x) - f (y) .

Hence, by defining

{\tilde{a}}_{j} = 1 + e^{2 π i ξ_{j}}

, for

j = 1, 2

, we obtain the following representation of the graph Laplacian as a matrix-valued multiplication operator.

Proposition 5.1 (Parra & Richard, 2018, Proposition 4.7)

There exists a unitary operator $\tilde{U} : ℓ (\tilde{V}) \to L^{2} (T^{2}; C^{n})$ such that $\tilde{U} ({\tilde{H}}_{0}) {\tilde{U}}^{*} = {\tilde{h}}_{0}$ where ${\tilde{h}}_{0}$ denotes the multiplication operator by the real analytic function ${\tilde{h}}_{0} : T^{2} \to M_{3 \times 3} (C)$ on $L^{2} (T^{2}, C^{3})$ given by the following equation:

{\tilde{h}}_{0} (ξ) = (\begin{matrix} 4 & - {\tilde{a}}_{1} (ξ) & - {\tilde{a}}_{2} (ξ) \\ - \bar{{\tilde{a}}_{1} (ξ)} & 2 & 0 \\ - \bar{{\tilde{a}}_{2} (ξ)} & 0 & 2 \end{matrix}) .

(30)

Setting as before $\tilde{r} (ξ) = | {\tilde{a}}_{1} (ξ) |^{2} + | {\tilde{a}}_{2} (ξ) |^{2}$ , and noticing

| {\tilde{a}}_{j} (ξ) |^{2} = 2 + 2 \cos (2 π ξ_{j}) = 4 \cos^{2} (π ξ_{j}),

(31)

we can obtain the associated characteristic polynomial to

{\tilde{h}}_{0}

{\tilde{p}}_{z} (ξ) = (2 - z) (z^{2} - 6 z + 8 - \tilde{r} (ξ)),

and the corresponding non-constant band functions

{\tilde{z}}_{\pm} (ξ) = 3 \pm \sqrt{1 + \tilde{r} (ξ)} .

It follows that the spectrum satisfies

σ ({\tilde{H}}_{0}) = σ_{ess} ({\tilde{H}}_{0}) = σ_{ac} ({\tilde{H}}_{0}) = [0, 2] ⋃ [4, 6],

(32)

with

2

an embedded degenerated eigenvalue. Given

\tilde{V} : \tilde{V} \to R

we define the Schrödinger operator

\tilde{H} = {\tilde{H}}_{0} + \tilde{V},

and the corresponding eigenvalue counting function by the following equation:

\tilde{N} (λ) = Rank 1_{(2 + λ, 3)} (H),

for

λ \in (0, 1)

. As before, by taking the limit

λ ↓ 0

, we will be able to study the accumulation of eigenvalues near the perturbed flat band.

Remark 5.2

An attentive reader can wonder why this Laplacian operator show the same spectral properties than the Dirac operator studied in previous sections (see the proof of Theorem 5.3). In general, one can say that the clear distinction of the order of a differential operator gets muddy in the discrete case, see, for instance, the discussion related to the continuum limit of discrete Dirac operators (Cornean et al., 2022; Nakamura, 2024).

5.2. Admissible Perturbations and Eigenvalue Asymptotics

Let us start by noticing that for every $μ \in Z^{2}$ we can define $f_{μ} \in ℓ^{2} (\tilde{V})$ by the following equation:

f_{μ} (x) = {\begin{cases} 1 & if x = μ + (\frac{1}{2}, 0), \\ - 1 & if x = μ + (0, \frac{1}{2}), \\ 0 & else, \end{cases}

and it satisfies

{\tilde{H}}_{0} f_{μ} = 2 f_{μ}

. Hence, if we decompose

ℓ^{2} (\tilde{V})

ℓ^{2} (\tilde{V}) ≅ ℓ^{2} (Z^{2}) \oplus ℓ^{2} (Z^{2} + (\frac{1}{2}, 0)) \oplus ℓ^{2} (Z^{2} + (0, \frac{1}{2})),

we have that

Ker ({\tilde{H}}_{0} - 2) \subset {0} \oplus ℓ^{2} (Z^{2} + (\frac{1}{2}, 0)) \oplus ℓ^{2} (Z^{2} + (0, \frac{1}{2})) .

Then, if we define

{\tilde{v}}_{j} : Z^{2} \to R

, for

j \in {0, 1, 2}

{\tilde{v}}_{0} (μ) = \tilde{V} (μ), {\tilde{v}}_{1} (μ) = \tilde{V} (μ + (\frac{1}{2}, 0)), and {\tilde{v}}_{2} (μ) = \tilde{V} (μ + (0, \frac{1}{2})),

we can apply Definition 3.1 to

\tilde{V}

Let us now observe that for any $z \notin σ ({\tilde{H}}_{0})$

\begin{aligned} ({\tilde{h}}_{0} - z)^{- 1} & = \frac{1}{p_{z}} (\begin{matrix} (2 - z)^{2} & {\tilde{a}}_{1} (2 - z) & {\tilde{a}}_{2} (2 - z) \\ \bar{{\tilde{a}}_{1}} (2 - z) & (2 - z) (4 - z) - | a_{2} |^{2} & \bar{{\tilde{a}}_{1}} {\tilde{a}}_{2} \\ \bar{{\tilde{a}}_{2}} (2 - z) & {\tilde{a}}_{1} \bar{{\tilde{a}}_{2}} & (2 - z) (4 - z) - | a_{1} |^{2} \end{matrix}) \\ = \frac{1}{(z^{2} - 6 z + 8 - \tilde{r})} (\begin{matrix} (2 - z) & {\tilde{a}}_{1} & {\tilde{a}}_{2} \\ \bar{{\tilde{a}}_{1}} & (4 - z) & 0 \\ \bar{{\tilde{a}}_{2}} & 0 & (4 - z) \end{matrix}) + \frac{1}{p_{z}} (\begin{matrix} 0 & 0 & 0 \\ 0 & - | a_{2} |^{2} & \bar{{\tilde{a}}_{1}} {\tilde{a}}_{2} \\ 0 & {\tilde{a}}_{1} \bar{{\tilde{a}}_{2}} & - | a_{1} |^{2} \end{matrix}) . \end{aligned}

Hence, we define

\tilde{M} : T^{2} \to M_{3 \times 3} (C)

by the following equation:

\tilde{M} := \frac{1}{\tilde{r}} (\begin{matrix} 0 & 0 & 0 \\ 0 & | {\tilde{a}}_{2} |^{2} & - {\tilde{a}}_{2} \bar{{\tilde{a}}_{1}} \\ 0 & - {\tilde{a}}_{1} \bar{{\tilde{a}}_{2}} & | {\tilde{a}}_{1} |^{2} \end{matrix}) .

(33)

Theorem 5.3

Assume that $\tilde{V}$ is an admissible perturbation of order $γ$ and associate $3 \times 3$ matrix $\tilde{Γ}$ . Define the constant $\tilde{C}$ by

\tilde{C} := \int_{T^{2}} Tr ((Γ \tilde{M} (ξ))^{\frac{n}{γ}}) d ξ .

(34)

Then, the eigenvalue counting function satisfies

\tilde{N} (λ) = λ^{- \frac{n}{γ}} (\tilde{C} τ_{n} + o (1)), λ ↓ 0.

(35)

Proof.

The proof follows the same path of the proof of Theorem 3.2 once the following considerations are taken into account: the symbol on $T^{2}$ of ${\tilde{H}}_{0} - 3$ correspond to the symbol of $H_{0}$ with $m = 1$ and replacing $a_{j}$ with $- {\tilde{a}}_{j}$ . Then, by comparing (31) with (9) we can see that the behavior of the non-constant band near $2$ for ${\tilde{h}}_{0}$ is the same as in the one for $h_{0}$ near $- 1 = - m$ . □

Footnotes

Acknowledgments

Both authors gratefully acknowledge the hospitality of the Institut de Mathématiques de Bordeaux were the final draft of this article was prepared.

ORCID iD

Daniel Parra

Funding

Pablo Miranda was supported by the Chilean Fondecyt Grant 1201857. Daniel Parra was supported by Universidad de La Frontera, Apoyo PF24-0027.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Appendix A. Formulas for Arbitrary n

In this appendix, we gather some straight-forward result concerning the symbol $h_{0} (ξ)$ for arbitrary $n$ for the convenience of the reader.

Lemma A.1. For any $n > 1$ , we have

\det (h_{0} - z) = (- 1)^{n} (m + z)^{n - 1} (m^{2} - z^{2} + \sum_{l = 1}^{n} | a_{l} |^{2}) .

Proof.

Let us prove this formula by induction. The case $n = 2$ is a direct computation. Next, we consider $n > 2$ and compute

\begin{aligned} | \begin{matrix} m - z & a_{1} & \dots & a_{n} \\ \bar{a_{1}} & - m - z & \dots & 0 \\ ⋮ & ⋱ & ⋮ \\ \bar{a_{n}} & 0 & \dots & - m - z \end{matrix} | \\ = (- 1)^{n} \bar{a_{n}} | \begin{matrix} a_{1} & \dots & a_{n - 1} & a_{n} \\ - m - z & \dots & 0 & 0 \\ ⋮ & \dots & ⋮ & ⋮ \\ 0 & \dots & - m - z & 0 \end{matrix} | + (- m - z) | \begin{matrix} m - z & a_{1} & \dots & a_{n - 1} \\ \bar{a_{1}} & - m - z & \dots & 0 \\ ⋮ & ⋱ & ⋮ \\ \bar{a_{n - 1}} & 0 & \dots & - m - z \end{matrix} | \\ = (- 1)^{n} (m + z)^{n - 1} | a_{n} |^{2} + (- 1)^{n} (m + z)^{n - 1} (m^{2} - z^{2} + \sum_{l = 1}^{n - 1} | a_{l} |^{2}) \\ = (- 1)^{n} (m + z)^{n - 1} (m^{2} - z^{2} + \sum_{l = 1}^{n} | a_{l} |^{2}) . ◼ \end{aligned}

□

Lemma A.2. Formula (10) is valid Proof.

It is enough to compute the product between the matrix part of the right-hand size of (10) and $h_{0} - z$ :

(\begin{matrix} (z + m)^{2} & a_{1} (z + m) & \dots & a_{n} (z + m) \\ \bar{a_{1}} (z + m) & z^{2} - m^{2} - r_{1} & \dots & a_{n} \bar{a_{1}} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ \bar{a_{n}} (z + m) & a_{1} \bar{a_{n}} & \dots & z^{2} - m^{2} - r_{n} \end{matrix}) (\begin{matrix} m - z & a_{1} & \dots & a_{n} \\ \bar{a_{1}} & - m - z & \dots & 0 \\ ⋮ & ⋱ & ⋮ \\ \bar{a_{n}} & 0 & \dots & - m - z \end{matrix}) .

Let us denote this product by

M := (m_{i j})_{0 ⩽ i, j ⩽ n}

. First we compute the diagonal terms

m_{i i}

. For

i = 0

, we compute

(\begin{matrix} (z + m)^{2} & a_{1} (z + m) & \dots & a_{n} (z + m) \end{matrix}) (\begin{matrix} m - z \\ \bar{a_{1}} \\ ⋮ \\ \bar{a_{n}} \end{matrix}) = (z + m) ((z + m) (m - z) + \sum_{l = 1}^{n} | a_{l} |^{2}),

while for

i > 0

, the product

(\begin{matrix} \bar{a_{i}} (z + m) & a_{1} \bar{a_{i}} & \dots & z^{2} - m^{2} - r_{i} & \dots & a_{n} \bar{a_{i}} \end{matrix}) (\begin{matrix} a_{i} \\ 0 \\ ⋮ \\ - m - z \\ ⋮ \\ 0 \end{matrix})

gives

| a_{i} |^{2} (z + m) - (z + m) (z^{2} - m^{2} - r_{i}) .

In both cases, we obtain

(m + z) (m^{2} - z^{2} + r)

We turn our attention to the off diagonal terms $m_{i j}$ with $i < j$ . For $i = 0$ , we get

(\begin{matrix} (z + m)^{2} & a_{1} (z + m) & \dots & a_{n} (z + m) \end{matrix}) (\begin{matrix} a_{j} \\ 0 \\ ⋮ \\ - m - z \\ ⋮ \\ 0 \end{matrix}) = a_{j} (z + m)^{2} + a_{j} (z + m) (- m - z),

while for

i > 0

, we get

(\begin{matrix} \bar{a_{i}} (z + m) & a_{1} \bar{a_{i}} & \dots & z^{2} - m^{2} - r_{i} & \dots & a_{n} \bar{a_{i}} \end{matrix}) (\begin{matrix} a_{j} \\ 0 \\ ⋮ \\ - m - z \\ ⋮ \\ 0 \end{matrix}) = \bar{a_{i}} (z + m) a_{j} + a_{j} \bar{a_{i}} (- m - z) .

In both cases, the result is equal to zero. By symmetry, we obtain the same results for the off diagonal terms with

j < i

. We conclude that

M = diag ((m + z) (m^{2} - z^{2} + r))

and hence (10) is valid. □

References

Anné

Torki-Hamza

(2015). The Gauss-Bonnet operator of an infinite graph. Analysis and Mathematical Physics, 5(2), 137–159. https://doi.org/10.1007/s13324-014-0090-0.

Bergholtz

E. J.

Liu

(2013). Topological flat band models and fractional Chern insulators. International Journal of Modern Physics B, 27(24), 1330017.

Birman

M. S.

Karadzhov

G. E.

Solomyak

M. Z.

(1991). Boundedness conditions and spectrum estimates for the operators

b (X) a (D)

and their analogs. In Estimates and asymptotics for discrete spectra of integral and differential equations (Leningrad, 1989–90), Adv. Soviet Math., Vol. 7, Amer. Math. Soc., Providence, RI. pp.85–106.

Birman

M. S.

Solomjak

M. Z.

(1970). Asymptotics of the spectrum of weakly polar integral operators. Izvestiya Akademii Nauk SSSR Series in Materials, 34, 1142–1158.

Birman

M. S.

Solomjak

M. Z.

(1987). Spectral theory of selfadjoint operators in Hilbert space. Mathematics and its Applications (Soviet Series). D. Reidel Publishing Co., Dordrecht, Translated from the 1980 Russian original by S. Khrushchëv and V. Peller.

Cornean

Garde

Jensen

(2022). Discrete approximations to Dirac operators and norm resolvent convergence. Journal of Spectral Theory, 12(4), 1589–1622.

Iwatsuka

Tamura

(1998). Asymptotic distribution of eigenvalues for Pauli operators with nonconstant magnetic fields. Duke Mathematical Journal , 93(3), 535–574.

Kerner

Täufer

Wintermayr

(2024). Robustness of flat bands on the perturbed Kagome and the perturbed Super-Kagome lattice. Annales Henri Poincaré. 25, 3831–3857. https://doi.org/10.1007/s00023-023-01399-7.

Klaus

(1982/83). Some applications of the Birman-Schwinger principle. Helvetica Physica Acta , 55(1), 49–68.

10.

Kollár

A. J.

Fitzpatrick

Sarnak

Houck

A. A.

(2020). Line-graph lattices: Euclidean and non-Euclidean flat bands, and implementations in circuit quantum electrodynamics. Communications in Mathematical Physics, 376(3), 1909–1956.

11.

Korotyaev

Saburova

(2014). Schrödinger operators on periodic discrete graphs. Journal of Mathematical Analysis and Applications, 420(1), 576–611.

12.

Kuchment

P. A.

(1991). On the floquet theory of periodic difference equations. In Geometrical and algebraical aspects in several complex variables (Cetraro, 1989), Sem. Conf., Vol. 8, EditEl, Rende, pp.201–209.

13.

Miranda

Parra

Raikov

(2023). Spectral asymptotics at thresholds for a Dirac-type operator on

Z^{2}

. Journal of Functional Analysis , 284(2), 109743.

14.

Nakamura

(2024). Remarks on discrete Dirac operators and their continuum limits. Journal of Spectral Theory, 14(1), 255–269. https://doi.org/10.4171/jst/478.

15.

Parra

(2017). Spectral and scattering theory for Gauss-Bonnet operators on perturbed topological crystals. Journal of Mathematical Analysis and Applications, 452(2), 792–813.

16.

Parra

Richard

(2018). Spectral and scattering theory for Schrödinger operators on perturbed topological crystals. Reviews in Mathematical Physics , 30(4), 1850009.

17.

Pushnitski

(2009). Operator theoretic methods for the eigenvalue counting function in spectral gaps. Annales Henri Poincaré, 10(4), 793–822.

18.

Pushnitski

Rozenblum

(2011). On the spectrum of Bargmann-Toeplitz operators with symbols of a variable sign. Journal d’Analyse Mathematique , 114, 317–340.

19.

Pushnitski

Sobolev

(2023). Hankel operators with band spectra and elliptic functions. arXiv preprint, arxiv.org/abs/2307.09242.

20.

Raikov

(1990). Eigenvalue asymptotics for the Schrödinger operator with homogeneous magnetic potential and decreasing electric potential. I. Behaviour near the essential spectrum tips. Communications in Partial Differential Equations, 15(3), 407–434.

21.

Reed

Simon

(1978). Methods of modern mathematical physics. IV. Analysis of operators. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London.

22.

Sabri

Youssef

(2023). Flat bands of periodic graphs. Journal of Mathematical Physics, 64(9), 092101.

23.

Sunada

(2013). Topological crystallography, Surveys and tutorials in the applied mathematical sciences, Vol. 6. Springer, Tokyo. With a view towards discrete geometric analysis. ISBN 978-4-431-54176-9; 978-4-431-54177-6. https://doi.org/10.1007/978-4-431-54177-6.

24.

Thaller

(1992). The Dirac equation. Texts and Monographs in Physics. Springer-Verlag, Berlin.

25.

Zworski

(2024). Mathematical results on the chiral models of twisted bilayer graphene. Journal of Spectral Theory, 14(3), 1063–1107.

Eigenvalue Asymptotics Near a Flat Band in the Presence of a Slowly Decaying Potential

Abstract

Keywords

1. Introduction

2.1. Discrete Dirac Operator

Proposition 2.1 Parra, 2017, Proposition 3.5

4.1. Some Notation and Results on Compact Operators

4.2. Effective Hamiltomian

5.1. A Simple Example of a Z 2 -Periodic Graph With a Flat Band

Footnotes

Acknowledgments

ORCID iD

Funding

Declaration of Conflicting Interests

Appendix A. Formulas for Arbitrary n

References

5.1. A Simple Example of a $Z^{2}$ -Periodic Graph With a Flat Band